Auto-correlation, often abbreviated as AC, is a statistical tool used to measure the degree of similarity between a given 時系列 and a lagged version of itself over successive time intervals. This concept is particularly important in 時系列分析, where understanding the intrinsic patterns and dependencies within the data can lead to better predictions and insights.
実用的には、自己相関は、系列の現在の値が過去の値と関連しているかどうかを識別するのに役立ちます。例えば、金融市場では、自己相関は株価がトレンドに従うか平均回帰の挙動を示すかどうかを示すことがあります。高い正の自己相関は、高い値が高い値に続きやすい（低い値も同様）ことを示し、負の自己相関は、高い値の後に低い値が続く傾向を示します。
自己相関の数学的表現は computing the 相関係数 between the time series and its lagged versions. The formula for the auto-correlation function (ACF) is given by:
ACF(k) = Cov(X_t, X_{t-k}) / (Var(X_t))
where X_t represents the time series values at time t, 共分散 is the covariance, 分散 is the variance, and k は遅延を示します。
自己相関は、経済学をはじめとするさまざまな分野で広く使用されており、 meteorology, and engineering, where understanding how current conditions relate to past conditions is crucial for forecasting and modeling. However, care must be taken to avoid misinterpretation, as auto-correlation can sometimes be influenced by external factors or trends present in the data.